For $f : \mathbb{ N} \to \{ 0,\pm 1\}$ the $f$-signed partition numbers $\mathfrak{ p}(n,f)$ are defined to be the weighted partition sums \[ \mathfrak{ p}(n,f) = \sum_{ \substack{ x_{ 1}+\cdots+x_{ k} = n \\ x_{ 1} \geq \cdots \geq x_{ k} > 0 \\ k \geq 1}} f(x_{ 1})f(x_{ 2})\cdots f(x_{ k}). \] For prime $p > 2$, let $(\frac{ \cdot}{ p})$ denote the Legendre symbol modulo $p$. The first half of this paper derives Rademacher-style series formulae for the quantities $\mathfrak{ p}(n,\pm(\frac{ \cdot}{ p}))$ for $p < 24$ satisfying $p \equiv 1 \pmod{ 4}$ (that is, for $p=5,13,17$), and the extensions to general $p \equiv 1 \pmod{ 4}$ are made apparent in our derivations. In the second half of this paper, the series formulae for $\mathfrak{ p}(n,\pm(\frac{ \cdot}{ 17}))$, as well as various properties of Dedekind sums and their "character-twisted" analogues, are used to establish that these two quantities are identically zero on certain (mod $34$)-arithmetic progressions.
Categories: math.NT